The linking von Neumann algebras of W*-TROs

TL;DR

The paper addresses when a von Neumann algebra ${\mathscr M}$ can be realized as the linking von Neumann algebra $R(V)$ of a $W^\ast$-TRO $V$, proving this holds precisely when ${\mathscr M}$ has no abelian direct summand. It provides a constructive correspondence via a projection $e$ with central covers $C_e=C_{I-e}=I$, showing $V=e{\mathscr M}(I-e)$ yields $R(V)={\mathscr M}$. Furthermore, it develops new characterizations of nuclear TROs and $W^\ast$-exact TROs entirely in terms of their corners $C(V)$, $D(V)$, and the linking algebra $A(V)$, including injectivity properties of $V^{**}$ and $A(V)^{**}$. These results extend prior work by Kaur and DR and yield practical criteria for injectivity and weak$^*$-exactness within the linking-algebra framework.

Abstract

In this note, we show that a von Neumann algebra can be written as the linking von Neumann algebra of a W*-TRO if and only if it contains no abelian direct summand. We also provide some new characterizations of nuclear TROs and $W^\ast$-exact TROs in terms of the properties of their linking algebras.

Theorems & Definitions (16)

• Theorem 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• Theorem 2.4
• Lemma 2.5: BO
• Proposition 2.6: DR
• Lemma 2.7: DR
• Lemma 2.8